gröbner geometry of schubert polynomials through ice
TRANSCRIPT
Grobner Geometry of Schubert Polynomials...
Through Ice
arXiv:2003.13719
Anna Weigandt
University of Michigan
AlCoVE: June 16, 2020
Based on joint work with Zachary Hamaker (Florida) and Oliver Pechenik (Michigan)
Table of contents
Schubert Varieties
Grobner Geometry
Bumpless Pipe Dreams
1
Schubert Varieties
Schubert Classes in Cohomology
The complete flag variety is the quotient F`(n) = GL(n)/B.
There’s a natural action of B on F`(n) by left multiplication. The
orbits Ωw are called Schubert cells and give rise to the Bruhat
decomposition:
F`(n) =∐w∈Sn
Ωw .
The Schubert varieties are the closures of these orbits: Xw = Ωw .
Schubert varieties give rise to Schubert classes σw in the
cohomology ring H∗(F`(n)). The Schubert classes form a linear
basis for H∗(F`(n)).
2
The Borel Isomorphism
Thanks to Borel, there is an isomorphism
Φ : H∗(F`(n))→ Z[x1, x2, . . . , xn]/I
where I is the ideal generated by the (non-constant) elementary
symmetric polynomials.
Question: What is a “good” polynomial representative for the
coset Φ(σw )?
One Answer: Schubert polynomials (Lascoux-Schutzenberger
1982).
3
The Definition of Sw (x)
Start with the longest permutation in Sn
w0 = n n − 1 . . . 1 Sw0(x) := xn−11 xn−22 . . . xn−1.
The rest are defined recursively by divided difference operators:
∂i f :=f − si · fxi − xi+1
and Swsi (x) := ∂iSw (x) if w(i) > w(i + 1).
There are also double Schubert polynomials, which are defined
by the same operators, with the initial condition
Sw0(x; y) :=∏
i+j≤n(xi − yj).
4
But how natural is this choice?
Monomial positivity:
S14523 = x21x22 + x21x2x3 + x21x
23 + x1x
22x3 + x1x2x
23 + x22x
23
Stability: For w ∈ Sn, Sw = Sι(w) where ι : Sn → Sn+1 is the
natural inclusion.
No cleanup:
Su ·Sv =∑w
cwu,vSw ⇒ σ ∪ σv =∑w
cwu,vσw
Lift of the Schur polynomials: Each Schur polynomial is itself a
Schubert polynomial.
5
Plus, there’s many combinatorial formulas
Theorem (Fomin-Kirillov 1996 / Bergeron-Billey 1993)
The double Schubert polynomial Sw (x; y) is the weighted sum
Sw (x; y) =∑
P∈Pipes(w)
wt(P).
Example:
S2143 = (x1− y1)(x3− y1) + (x1− y1)(x2− y2) + (x1− y1)(x1− y3).
6
Yes, but what about geometric naturality?
Degeneracy Loci: Fulton (1992) expressed Chern class formulas
for degeneracy loci of maps of flagged vector bundles in terms of
double Schubert polynomials.
Grobner Geometry: Knutson and Miller (2005) used a specific
geometric setup and explicit Grobner degenerations to uniquely
identify Schubert polynomials as the “right” representatives for
Schubert classes.
7
Grobner Geometry
Matrix Schubert Varieties
There’s a natural projection
π : GL(n)→ F`(n)
and inclusion
ι : GL(n)→ Mat(n).
Using these, Fulton (1982) defined the matrix Schubert variety:
Xw := ι (π−1(Xw )).
Fulton also introduced the Schubert determinantal ideal Iw , and
gave explicit generators. Fulton showed Iw is prime and
Xw = V (Iw ).
8
Roughly, Xw ⊆ Mat(n) is defined by rank conditions on maximal
northwest submatrices.
Example: For w = 2143, the matrix Schubert variety Xw looks
like this:
(mij) ∈ Mat(4) : rk(m11) = 0 and rk
m11 m12 m13
m21 m22 m23
m31 m32 m33
≤ 2.
It is cut out by the Schubert determinantal ideal:
Iw =
⟨z11,
∣∣∣∣∣∣∣z11 z12 z13
z21 z22 z23
z31 z32 z33
∣∣∣∣∣∣∣⟩.
9
Multidegrees
The group T× T acts on the space of n × n matrices Mat(n) by
(t, u) ·M := tMu−1
and endows its coordinate ring with a Z2n grading.
The multidegree C(M; x; y) is a function from Z2n graded modules
M to polynomials in Z[x1, . . . , xn; y1, . . . , yn].
Whenever a subvariety X ⊆ Mat(n) is stable under the action of
T× T, its coordinate ring is a Z2n graded module. In this case, we
write C(X ; x; y).
Theorem (Knutson-Miller 2005)
C(Xw ; x; y) = Sw (x; y).
10
Computing C(X ; x; y)
Let LD be the coordinate subspace defined by setting the
coordinate zij = 0 whenever (i , j) ∈ D. By normalization
C(LD ; x; y) =∏
(i ,j)∈D
(xi − yj).
By additivity, if X =⋃n
i=1 Xi , is a (possibly scheme theoretic)
union then
C(X ; x; y) =∑i∈I
multXi(X )C (Xi ; x; y),
where the sum is over Xi so that codim(Xi ) = codim(X ).
Finally, multidegrees are preserved by “nice enough”
degenerations.
11
Grobner Degenerations
Let Z = (zij) be a matrix of generic variables. Fix a monomial
term order ≺ on C[Z ]. Write init≺(f ) for the lead term of f .
The initial ideal of I is init≺(I ) = 〈init≺(f ) : f ∈ I 〉.
If G ⊆ I and init≺(I ) = 〈init≺(g) : g ∈ G 〉 then G is a Grobner
basis for I .
It’s a fact that finite Grobner bases always exist and Buchberger’s
algorithm gives an explict way to find them.
12
A Toy Example
Take the Schubert determinantal ideal
I2143 =
⟨z11,
∣∣∣∣∣∣∣z11 z12 z13
z21 z22 z23
z31 z32 z33
∣∣∣∣∣∣∣⟩
and degenerate with respect to an antidiagonal term order.
init≺a(I2143) = 〈z11, z13z22z31〉 = 〈z11, z31〉∩ 〈z11, z22〉∩ 〈z11, z13〉.
13
Pipe dreams are natural
Theorem (Knutson-Miller 2005)
1. With respect to any antidiagonal term order, Fulton’s
generators for Iw are a Grobner basis.
2. The initial scheme of Xw with respect to an antidiagonal term
order is a union of coordinate subspaces indexed by Pipes(w).
init≺a(Xw ) =⋃
P∈Pipes(w)
LC(P).
In particular,
Sw (x; y) =∑
P∈Pipes(w)
C(LC(P); x; y) =∑
P∈Pipes(w)
wt(P).
14
What about diagonal term orders?
Knutson-Miller-Yong (2005) showed Fulton’s generators are
Grobner for a diagonal term order if and only if w is vexillary (2143
avoiding).
KMY also showed how to label the components of degenerations
of vexillary matrix Schubert varieties with flagged tableaux and
diagonal pipe dreams.
Outside of the vexillary setting, limits might fail to be reduced.
This was discouraging.
15
Bumpless Pipe Dreams
Bumpless Pipe Dreams
A bumpless pipe dream is a tiling of the n × n grid with the six
tiles pictured above so that there are n pipes which
1. start at the right edge of the grid,
2. end at the bottom edge, and
3. pairwise cross at most one time.
The diagram of a BPD, denoted D(P), is the set of blank tiles.
16
Bumpless Pipe Dreams
Example Not an Example
17
The Permutation of a Bumpless Pipe Dream
1 2 3 4 5
1
4
3
2
5
Write BPipes(w) for the set of bumpless pipe dreams which trace
out the permutation w .
18
Theorem (Lam-Lee-Shimozono 2018)
The double Schubert polynomial Sw (x; y) is the weighted sum
Sw (x; y) =∑
P∈BPipes(w)
wt(P).
S2143 = (x1− y1)(x3− y3) + (x1− y1)(x2− y1) + (x1− y1)(x1− y2).
19
Example: w = 2143
Sw = (x1 − y1)(x3 − y3) + (x1 − y1)(x2 − y1) + (x1 − y1)(x1 − y2).
Sw = (x1 − y1)(x3 − y1) + (x1 − y1)(x2 − y2) + (x1 − y1)(x1 − y3).
20
Lascoux’s ASM Formula
Lascoux (2002) gave a formula for double Grothendieck
polynomials as a weighted sum over alternating sign matrices.
You can reformulate Lascoux’s work in terms of (possibly
non-reduced) bumpless pipe dreams (W- 2020).
Using this perspective, you can recover the formula of LLS.
21
Why Ice?
22
From vexillary BPD to tableaux
Theorem (W- 2020)
There’s a bijection from vexillary BPDs to flagged tableaux.
2 2
3
1 2
3
1 2
2
1 1
3
1 1
2
Is it possible BPDs are the “right” combinatorial object to describe
diagonal Grobner degenerations?
23
Take the Schubert determinantal ideal
I2143 =
⟨z11,
∣∣∣∣∣∣∣z11 z12 z13
z21 z22 z23
z31 z32 z33
∣∣∣∣∣∣∣⟩
and degenerate with respect to a diagonal term order.
init≺d(I2143) = 〈z11, z12z21z33〉 = 〈z11, z33〉∩〈z11, z21〉∩〈z11, z12〉.
24
Permutations with Duplicate BPD Diagrams
Write Dup(n) for the set of permutations in Sn which have
multiple BPDs with the same diagram.
Dup(6) = 214365, 321654.
25
A Conjecture
Conjecture (Hamaker-Pechenik-W- 2020)
For all (set theoretic) components of init≺d(Xw ),
multLD (init≺d(Xw )) = #P ∈ BPipes(w) : D(P) = D.
This would mean bumpless pipe dreams label irreducible
components of of the diagonal Grobner degeneration of Xw with
the correct multiplicity.
26
Different Generators
Take the “obvious” defining equations for Iw . Some of them may
be single variables zij . Throw away all other terms that contain
these variables. These are the CDG generators (Conca-De
Negri-Gorla 2015).
Fulton Generators:
I2143 =
⟨z11,
∣∣∣∣∣∣∣z11 z12 z13
z21 z22 z23
z31 z32 z33
∣∣∣∣∣∣∣⟩
CDG Generators:
I2143 = 〈z11,−z12z21z33 + z12z23z31 + z13z21z32 − z13z22z31〉
We call w CDG if its CDG generators are a diagonal Grobner basis
for Iw .27
Another Class of Permutations
A permutation is predominant if its Lehmer code is of the form
λ0m` for some partition λ and m, ` ∈ N.
Theorem (Hamaker-Pechenik-W- 2020)
If w is predominant, then in≺d(Xw ) is reduced and CDG and the
main conjecture holds for w .
w = 42153:
.28
A block sum trick
Let u =
0 1 0
1 0 0
0 0 0
, v =
[1 0
0 0
], and w be the permutation
associated to the partial permutation u v . The permutation
matrix for w is
0 0 0 1 0 0
0 0 0 0 0 1
0 1 0 0 0 0
1 0 0 0 0 0
0 0 0 0 1 0
0 0 1 0 0 0
.
29
Block sums of BPDs
=
LemmaIf w = u v , then there is a bijection from BPipes(u)×BPipes(v)
to BPipes(w).
30
Main Theorem
We say a permutation is banner if it is a block sum of
predominant, copredominant, and vexillary partial permutations.
Theorem (Hamaker-Pechenik-W- 2020)
Let w be a banner permutation. Then
1. w is CDG, and
2. init≺d(Iw ) is radical; in particular
init≺d(Xw ) =
⋃P∈BPipes(w)
LD(P).
31
References i
Bergeron, N. and Billey, S. (1993). RC-graphs and Schubert
polynomials.
Experimental Mathematics, 2(4):257–269.
Fomin, S. and Kirillov, A. N. (1996). The Yang-Baxter
equation, symmetric functions, and Schubert polynomials.
Discrete Mathematics, 153(1):123–143.
Fulton, W. (1992). Flags, Schubert polynomials, degeneracy
loci, and determinantal formulas.
Duke Math. J, 65(3):381–420.
References ii
Hamaker, Z., Pechenik, O., and Weigandt, A. (2020). Grobner
geometry of Schubert polynomials through ice.
arXiv:2003.13719.
Knutson, A., Miller, E., and Yong, A. (2009). Grobner geometry
of vertex decompositions and of flagged tableaux.
Journal fur die reine und angewandte Mathematik (Crelles
Journal), 2009(630):1–31.
Lam, T., Lee, S. J., and Shimozono, M. (2018). Back stable
Schubert calculus.
arXiv:1806.11233.
Lascoux, A. (2002). Chern and Yang through ice.
References iii
Lascoux, A. and Schutzenberger, M.-P. (1982). Polynomes de
Schubert.
CR Acad. Sci. Paris Ser. I Math, 295(3):447–450.
Weigandt, A. (2020). Bumpless pipe dreams and alternating
sign matrices.
arXiv:2003.07342.
Questions?